1. **State the problem:** Calculate the value of the expression $\frac{1}{3} \times 3.14 \times 4^2 \times 6$. 2. **Recall the order of operations:** We first evaluate exponents, then multiplication and division from left to right. 3. **Evaluate the exponent:** $$4^2 = 16$$ 4. **Substitute back into the expression:** $$\frac{1}{3} \times 3.14 \times 16 \times 6$$ 5. **Multiply the numbers step-by-step:** $$3.14 \times 16 = 50.24$$ 6. **Now multiply by 6:** $$50.24 \times 6 = 301.44$$ 7. **Multiply by $\frac{1}{3}$:** $$\frac{1}{3} \times 301.44 = \frac{\cancel{1}}{\cancel{3}} \times 301.44 = \frac{301.44}{3} = 100.48$$ **Final answer:** $$100.48$$